Sir Wint1Am Rowan Hamitton’s Researches respecting Quaternions. 233 
x Gp Sle XE class 
Xi Xo =f Xot Sl X, = 4X3 
Xo Xi =SXotS Xi =4Xi3 (173) 
SG Sh eS 
=a'(e'd—cd')x,+ (e+ a) x3 
and these equations are, as we aimed that they should be, independent of the 
original derivand, that is, here, of the ordinal couple (a,,a,). In fact, the three 
first equations (173) are evidently true, by (172), whatever the constant co- 
efficients of derivation included in the sign x, may be; and if, by the definition 
(162) of that sign of derivation, we form the successive derivative, 
X1 X1 (aos a) = X1 (4, a,,,) 
= (ca, 0 + ca, Vv da,, as da, ) 
= (e(ca, + c'a,) + ¢'(da, + d’a,), d(ca, + c’a,) + d’(da,+d’a,)), (174) 
we are conducted, whatever the two original constituent ordinal relations a, and 
a, may be, to the same final ordinal couple, as if we add together the two partial 
results, which are obtained by the two derivations represented by the two terms 
of the last member of the fourth equation (173), namely, the two following 
couples : 
a~' (ce'd—cd') X, (a> a,) = ((e’'d—ed’) a, (c’d—cd')a,); } (175) 
(c +d’) X, (a a,) = ((e +’) (ca, c'a,), (e+ d’) (da,+ d’a,)). ‘ 
We may therefore express the result of two successive and complex derivations of 
this sort, performed on an ordinal couple (a,, a,), by a symbolical equation inde- 
pendent of that original derivand, or operand couple, namely, by the following : 
(M5 Xo + M X1) (MXo+ mM, X,) = mM Ky + my Xp (176) 
which is included in the form (152), and in which we have now these two rela- 
tions, of the form (149), between the numerical coefficients : 
mi, = am,m, + a*(c'd — ed’) mim, ; | (177) 
my = am, m, + am, m, + (e+ d’) mim. 
Under the same conditions we may also write, more briefly, 
(ms, mi") = (mM, M1) (AN ™,)s (178) 
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